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Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

math.toronto.edu

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Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#71

Post by mattlutze »

batbomb wrote: post #41
mightybyte wrote: post #35

I couldn't agree with the last point more. > A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives... I…

I'm not sure what that problem would have been, but an integral part (no pun intended) of DEs is expansions. It's not part of a bag of tricks, it's a very common technique used to solve a problem.

You know that from study and experience. If that underlying concept wasn't the lesson, like the parent indicated, it would I deep end up a trick in the bag.

The toughest part of teaching, I think, must be really knowing that your students are internalizing the core rules/principles/concepts behind the examples you teach with.

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#72

Post by xxs »

pkrumins wrote: post #6

One lesson academics should learn: pdf-naming-skills.pdf. I've been collecting interesting scientific papers and publications since early 2000 (I've a collection of 10,000 or so) and I've not yet seen a single academic, not even a computer scientist, who understands how to name your documents right so that when I download them I could quickly find them. I've to rename every single pdf. It's infuriating. Someone shoul…

When you download you have the option to pick a name (and not use the data provided by content-disposition)

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#73

Post by bitL »

mynegation wrote: post #27

My major is computational math, from 15 years ago, from leading Russian university, so it is just anecdata, and by no means should be generalized. I absolutely love mathematics, for me it is the embodiment of pure beauty. Still, I positively, absolutely hated the sophomore course of ODEs. The way it was taught was extremely abstract: here is the equation, this is integration, this is separation, this is your SLP, now…

I hated ODEs until I took MIT's robotics course and had to solve non-linear ones using non-linear optimization techniques. Since then I love them ;-)

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#74

Post by drjesusphd »

> I do not know how to properly motivate the Laplace transform

I feel like this is impossible without going to the complex plane. Like the author said, taking the inverse Laplace transform is no joke.

I feel like I never properly understood the Laplace transform until I learned about Landau damping. This is when waves exist, but are damped in a collisionless plasma. This damping is not disspiation and the energy does not get converted into heat. The usual way of presenting this is to show that if one Fourier transforms in time, you get the wrong answer. The fact that the system begins at a certain state, and is thus an initial value problem, needs to be respected.

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#75

Post by Terribledactyl »

tomekowal wrote: post #64

Please, write another textbook. The internet changed drastically the process of writing books. I saw people making profit from books available online for free. I saw books written chapter by chapter with errors found quickly by first readers. I saw systems that allow commenting parts that are not clear enough with comments how to clarify them. If you promise to deliver a textbook that teaches skills relevant to engin…

Rota died almost 17 years ago, I'm sorry.

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#76

Post by drjesusphd »

pkrumins wrote: post #12
GFK_of_xmaspast wrote: post #10

Earlier quoted context omitted.

Everybody has a different system, and why should they standardize on yours?

What kind of a system is surname.pdf, New-Pub_New2new.pdf, USENIX_.PDF, and Paper.pdf. I like to say: Show me how you organize your files, and I'll tell you how good of a computer user you are.

surname.pdf is almost certainly a half-hearted (because that is all that is possible) way to fit into someone else's naming scheme. Otherwise, it would be cv.pdf, application.pdf, etc. because those are the file names that make sense on my computer.

What you don't see is the directory structure that that file lives in, which gives almost all the necessary information.

~/work/projects/reinventwheel/paper/manuscript.pdf should and does contain more information than "manuscript.pdf". If you're not renaming files when you get them and putting them in the correct context on your machine, I would say it is you that is doing it wrong.

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#77

Post by semi-extrinsic »

drjesusphd wrote: post #74

> I do not know how to properly motivate the Laplace transform I feel like this is impossible without going to the complex plane. Like the author said, taking the inverse Laplace transform is no joke. I feel like I never properly understood the Laplace transform until I learned about Landau damping. This is when waves exist, but are damped in a collisionless plasma. This damping is not disspiation and the energy does…

The Laplace transform is somehow the continuous analog of a Taylor series expansion. You don't need complex analysis to motivate it. I sketched this for my students when TAing once upon a time, heavily inspired by this [1] nice MIT lecture.

I'm very surprised this isn't standard material. It makes the parallel between Laplace and Fourier transforms so much more intuitive, because you get Taylor series as a parallel to Fourier series.

[1] https://youtu.be/zvbdoSeGAgI

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#78

Post by amelius »

gohrt wrote: post #69
batbomb wrote: post #41

Earlier quoted context omitted.

I'm not sure what that problem would have been, but an integral part (no pun intended) of DEs is expansions. It's not part of a bag of tricks, it's a very common technique used to solve a problem.

The common techniques ARE a bag of tricks. Feynman was famous for being really good at integrals, because he had memorized the huge bag of tricks. Today, we have Mathematica for that, you don't need to be Feynman.

This makes me wonder. Is Mathematica also applying a bag of tricks (I suppose in a breadth first search), or does it have a more structural approach?

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#79

Post by agumonkey »

mightybyte wrote: post #35

I couldn't agree with the last point more. > A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality...In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives... I…

I don't know how correct this is http://www.math.harvard.edu/~knill/pedagogy/pechakucha/ but it makes the whole pre-college math education looks like a bag of trick. Here principles are both concrete and abstracted yet it all blends as one.

Re: Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

#80

Post by agumonkey »

mynegation wrote: post #27

My major is computational math, from 15 years ago, from leading Russian university, so it is just anecdata, and by no means should be generalized. I absolutely love mathematics, for me it is the embodiment of pure beauty. Still, I positively, absolutely hated the sophomore course of ODEs. The way it was taught was extremely abstract: here is the equation, this is integration, this is separation, this is your SLP, now…

I wonder if there are people that really click and appreciate purely abstract definitions of mathematical concepts. Many books and courses (especially in Europe it seems, watching U.S.A classes, Strang's for instance, feel a lot more pragmatic and applied). Sure after a few years of confusion one might finally click and realize all that was hidden behind a lemma, an identity or a symbol, and then that definition goes from painful to beautiful.
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